This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 52

2011 Denmark MO - Mohr Contest, 5
Tags: digit , last digit , perfect square , number theory
Determine all sets $(a, b, c)$ of positive integers where one obtains $b^2$ by removing the last digit in $c^2$ and one obtains $a^2$ by removing the last digit in $b^2$. .

2009 Puerto Rico Team Selection Test, 2
Tags: number theory , digit , perfect square , last digit
The last three digits of $ N$ are $ x25$. For how many values of $ x$ can $ N$ be the square of an integer?

2018 Junior Regional Olympiad - FBH, 4
Tags: last digit , digit , number theory
Determine the last digit of number $18^1+18^2+...+18^{19}+18^{20}$

2014 Lithuania Team Selection Test, 4
Tags: last digit , number theory , sum of digits
(a) Is there a natural number $n$ such that the number $2^n$ has last digit $6$ and the sum of the other digits is $2$? b) Are there natural numbers $a$ and $m\ge 3$ such that the number $a^m$ has last digit $6$ and the sum of the other digits is 3?

1983 All Soviet Union Mathematical Olympiad, 356
Tags: sequence , number theory , periodical , last digit , digit , floor function
The sequences $a_n$ and $b_n$ members are the last digits of $[\sqrt{10}^n]$ and $[\sqrt{2}^n]$ respectively (here $[ ...]$ denotes the whole part of a number). Are those sequences periodical?

1962 Dutch Mathematical Olympiad, 4
Tags: floor function , number theory , digit , last digit
Write using with the floor function: the last, the second last, and the first digit of the number $n$ written in the decimal system.

2000 Rioplatense Mathematical Olympiad, Level 3, 1
Tags: number theory , odd , multiple , last digit
Let $a$ and $b$ be positive integers such that the number $b^2 + (b +1)^2 +...+ (b + a)^2-3$ is multiple of $5$ and $a + b$ is odd. Calculate the digit of the units of the number $a + b$ written in decimal notation.

2012 Abels Math Contest (Norwegian MO) Final, 3a
Tags: number theory , product , factorial , last digit
Find the last three digits in the product $1 \cdot 3\cdot 5\cdot 7 \cdot . . . \cdot 2009 \cdot 2011$.

2019 Durer Math Competition Finals, 12
Tags: number theory , polynomial , last digit , digit
$P$ and $Q$ are two different non-constant polynomials such that $P(Q(x)) = P(x)Q(x)$ and $P(1) = P(-1) = 2019$. What are the last four digits of $Q(P(-1))$?

1996 Tournament Of Towns, (512) 5
Tags: number theory , last digit , digit
Does there exist a $6$-digit number $A$ such that none of its $500 000$ multiples $A$, $2A$, $3A$, ..., $500 000A$ ends in $6$ identical digits? (S Tokarev)

1961 Poland - Second Round, 4
Tags: number theory , last digit
Find the last four digits of $5^{5555}$.

1990 Greece National Olympiad, 4
Tags: number theory , last digit , digit
Since this is the $6$th Greek Math Olympiad and the year is $1989$, can you find the last two digits of $6^{1989}$?

1987 Tournament Of Towns, (150) 1
Tags: number theory , digit , last digit , power of 3
Prove that the second last digit of each power of three is even . (V . I . Plachkos)

2017 Finnish National High School Mathematics Comp, 3
Tags: last digit , number theory , digit
Consider positive integers $m$ and $n$ for which $m> n$ and the number $22 220 038^m-22 220 038^n$ has are eight zeros at the end. Show that $n> 7$.

1974 Swedish Mathematical Competition, 3
Tags: number theory , recurrence relation , last digit , digit
Let $a_1=1$, $a_2=2^{a_1}$, $a_3=3^{a_2}$, $a_4=4^{a_3}$, $\dots$, $a_9 = 9^{a_8}$. Find the last two digits of $a_9$.

1949-56 Chisinau City MO, 5
Tags: number theory , perfect square , last digit
Prove that the square of any integer cannot end with two fives.

2007 Denmark MO - Mohr Contest, 2
Tags: number theory , last digit
What is the last digit in the number $2007^{2007}$?

2006 Cuba MO, 8
Tags: last digit , power of 2 , number theory , digit
Prove that for any integer $k$ ($k \ge 2$) there exists a power of $2$ that among its last $k$ digits, the nines constitute no less than half. For example, for $k = 2$ and $k = 3$ we have the powers $2^{12} = ... 96$ and $2^{53} = ... 992$. [hide=original wording] Probar que para cualquier k entero existe una potencia de 2 que entre sus ultimos k dıgitos, los nueves constituyen no menos de la mitad. [/hide]

1988 Nordic, 1
Tags: last digit , number theory
The positive integer $ n$ has the following property: if the three last digits of $n$ are removed, the number $\sqrt[3]{n}$ remains. Find $n$.

2019 Polish Junior MO First Round, 1
Tags: number theory , last digit , digit
The natural number $n$ was multiplied by $3$, resulting in the number $999^{1000}$. Find the unity digit of $n$.

1979 Bundeswettbewerb Mathematik, 4
Tags: sequence , infinitely many solutions , number theory , last digit
An infinite sequence $p_1, p_2, p_3, \ldots$ of natural numbers in the decimal system has the following property: For every $i \in \mathbb{N}$ the last digit of $p_{i+1}$ is different from $9$, and by omitting this digit one obtains number $p_i$. Prove that this sequence contains infinitely many composite numbers.

1949-56 Chisinau City MO, 2
Tags: number theory , last digit
What is the last digit of $777^{777}$?

2001 Denmark MO - Mohr Contest, 2
Tags: number theory , factorial , digit , last digit
If there is a natural number $n$ such that the number $n!$ has exactly $11$ zeros at the end? (With $n!$ is denoted the number $1\cdot 2\cdot 3 \cdot ... (n - )1 \cdot n$).

2015 Puerto Rico Team Selection Test, 4
Tags: number theory , last digit
Let $n$ be a positive integer. Find as many as possible zeros as last digits the following expression: $1^n + 2^n + 3^n + 4^n$.

1952 Polish MO Finals, 5
Tags: number theory , last digit
Prove that none of the digits $2$, $4$, $7$, $9$ can be the last digit of a number $$ 1 + 2 + 3 + \ldots + n,$$ where $n$ is a natural number.